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Mirrors > Home > MPE Home > Th. List > clmnegsubdi2 | Structured version Visualization version GIF version |
Description: Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.) |
Ref | Expression |
---|---|
clmpm1dir.v | ⊢ 𝑉 = (Base‘𝑊) |
clmpm1dir.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmpm1dir.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
clmnegsubdi2 | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1136 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
2 | eqid 2731 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2731 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | 2, 3 | clmneg1 24527 | . . . 4 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊))) |
5 | 4 | 3ad2ant1 1133 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
6 | simp2 1137 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
7 | simpl 483 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
8 | 4 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
9 | simpr 485 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
10 | clmpm1dir.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | clmpm1dir.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
12 | 10, 2, 11, 3 | clmvscl 24533 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉) |
13 | 7, 8, 9, 12 | syl3anc 1371 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉) |
14 | 13 | 3adant2 1131 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉) |
15 | clmpm1dir.a | . . . 4 ⊢ + = (+g‘𝑊) | |
16 | 10, 2, 11, 3, 15 | clmvsdi 24537 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (-1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ (-1 · 𝐵) ∈ 𝑉)) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵)))) |
17 | 1, 5, 6, 14, 16 | syl13anc 1372 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵)))) |
18 | 10, 11, 15 | clmnegneg 24549 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵) |
19 | 18 | 3adant2 1131 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵) |
20 | 19 | oveq2d 7409 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + (-1 · (-1 · 𝐵))) = ((-1 · 𝐴) + 𝐵)) |
21 | clmabl 24514 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel) | |
22 | 21 | 3ad2ant1 1133 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ Abel) |
23 | simpl 483 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
24 | 4 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
25 | simpr 485 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
26 | 10, 2, 11, 3 | clmvscl 24533 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉) |
27 | 23, 24, 25, 26 | syl3anc 1371 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉) |
28 | 27 | 3adant3 1132 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉) |
29 | simp3 1138 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
30 | 10, 15 | ablcom 19631 | . . 3 ⊢ ((𝑊 ∈ Abel ∧ (-1 · 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴))) |
31 | 22, 28, 29, 30 | syl3anc 1371 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴))) |
32 | 17, 20, 31 | 3eqtrd 2775 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 1c1 11093 -cneg 11427 Basecbs 17126 +gcplusg 17179 Scalarcsca 17182 ·𝑠 cvsca 17183 Abelcabl 19613 ℂModcclm 24507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-seq 13949 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-mulg 18923 df-subg 18975 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-subrg 20310 df-lmod 20422 df-cnfld 20879 df-clm 24508 |
This theorem is referenced by: (None) |
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